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Voluntary disclosure, moral hazard and default risk * Shiming Fu 1 and Giulio Trigilia 2 1 Shanghai University of Finance and Economics 2 University of Rochester, Simon Business School June 19, 2019 Abstract We introduce voluntary disclosure opportunities in a dynamic agency model with non-verifiable cash flows. Disclosure has two countervailing effects. When bad news arise, the firm’s management is given some slack by the investors, which lowers the optimal pay-for-performance sensitivity. However, paying for bad luck also reduces the value for investors of providing liquidity to the firm in the first place. Disclosure lowers the firm’s probability of default conditional on a given performance history: it is associated with lower leverage and higher dividend payout rates. However, its effects in primary markets are heterogeneous across firms. At some low profitability firms, more frequent disclosure is associated to a higher probability of default and lower managerial pay. At other firms, both relations are reversed. Firms with intermediate performance history benefit the most from increasing the arrival rate of information to disclose, and are expected to disclose more frequently. Key words: voluntary disclosure, credit spreads, default risk, dynamic moral hazard, funding liquidity JEL classification: G32, D86, D61 * Previously circulated under the title: “Voluntary disclosure under dynamic moral hazard”. We thank Dan Bernhardt, Michael Gofman, Ron Kaniel, Meg Meyer, Alan Moreira, Dmitry Orlov, Heikki Rantakari, Clifford Smith, Vish Viswanathan, Ming Yang, Adriano Rampini, Martin Szydlowski (dis- cussant), Pavel Zruymov and the participants at: FTG (Kellogg), SFS Cavalcade (CMU), UNC Junior Finance Roundtable, Oxford (Nuffield College), CRETA (Warwick), Rochester (Simon), FTG Summer (LBS), SED (Mexico City), NASMES (Davis), MWET (Drexel). All errors are our own.

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Page 1: Voluntary disclosure, moral hazard and default risksof.shufe.edu.cn/_upload/article/files/.../4369f14c... · Voluntary disclosure, moral hazard and default risk Shiming Fu1 and Giulio

Voluntary disclosure, moral hazard and default risk ∗

Shiming Fu1 and Giulio Trigilia2

1Shanghai University of Finance and Economics2University of Rochester, Simon Business School

June 19, 2019

Abstract

We introduce voluntary disclosure opportunities in a dynamic agency model

with non-verifiable cash flows. Disclosure has two countervailing effects. When

bad news arise, the firm’s management is given some slack by the investors, which

lowers the optimal pay-for-performance sensitivity. However, paying for bad luck

also reduces the value for investors of providing liquidity to the firm in the first

place. Disclosure lowers the firm’s probability of default conditional on a given

performance history: it is associated with lower leverage and higher dividend payout

rates. However, its effects in primary markets are heterogeneous across firms. At

some low profitability firms, more frequent disclosure is associated to a higher

probability of default and lower managerial pay. At other firms, both relations

are reversed. Firms with intermediate performance history benefit the most from

increasing the arrival rate of information to disclose, and are expected to disclose

more frequently.

Key words: voluntary disclosure, credit spreads, default risk, dynamic moral

hazard, funding liquidity

JEL classification: G32, D86, D61

∗Previously circulated under the title: “Voluntary disclosure under dynamic moral hazard”. Wethank Dan Bernhardt, Michael Gofman, Ron Kaniel, Meg Meyer, Alan Moreira, Dmitry Orlov, HeikkiRantakari, Clifford Smith, Vish Viswanathan, Ming Yang, Adriano Rampini, Martin Szydlowski (dis-cussant), Pavel Zruymov and the participants at: FTG (Kellogg), SFS Cavalcade (CMU), UNC JuniorFinance Roundtable, Oxford (Nuffield College), CRETA (Warwick), Rochester (Simon), FTG Summer(LBS), SED (Mexico City), NASMES (Davis), MWET (Drexel). All errors are our own.

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1 Introduction

Over the last two decades, technological progress drastically reduced the costs of generat-

ing, storing and analyzing information. According to Dresner Advisory Services (2018),

which surveys 5,000 firms globally, adopters of the latest such technologies – broadly

referred to as Big Data and Artificial Intelligence – surged from 17% in 2015 to 59% in

2018, while another 30% signaled the intention to adopt them in the future.1 The most

frequent applications are data optimization and forecasting, which means that adopting

firms have increased their access to evidence that leads performance and are expected to

disclose information to their investors more frequently. So, we ask: what are the disclo-

sure patterns induced by these information technologies? And how does their adoption

affect firms’ default risk, pay-for-performance sensitivity, leverage and dividend payout?

Our paper highlights the difference between information generated by costly third-

party monitoring of a firm’s management, and information produced by a technology that

the firm’s management itself can control. This distinction is key. In practice, arm’s length

investors in a large firm have both limited incentives and ability to monitor, relying on the

firm itself to provide basic information about the sources of its performance. Conversely,

a firm’s management has access to information technologies that can sometimes shed light

on its likely near-term future performance, indirectly conveying information to outside

investors about how well the management itself is doing. The problem, of course, is that

while this information may not be easily manipulable, it can be shrouded by managers,

who have the discretion to not disclose information. The issue from the perspective of

the outside investors is that no disclosure could be due either to the absence of evidence,

or to its strategic concealment.

It is immediate that, if they can, a firm’s investors should design correct incentives for

managers to reveal information.2 We show how optimal incentive design differs when the

goal is to provide management incentives to reveal information, both good and bad, when

management receives it, versus the optimal design when monitoring is external to the firm

and costly. While in a typical monitoring setting the information is used by investors to

curb managerial rents,3 this need not be true in a disclosure setting. Especially at high

profitability firms, disclosure opportunities can be accompanied by higher rents paid to

management, which is optimal because rewarding disclosure lowers the deadweight losses

1Despite the increase in adoption, there still is a variety of competing infrastructures, most of whichare open-source. The most popular to this date are Spark, Kafka, MapReduce, Kubernetes and Yarn.

2This is an instance of Homlstrom’s informativeness principle (Holmstrom (1979)): investors candesign a randomization procedure that implements the allocation that was optimal with lower evidence,committing to disregard managerial disclosures probabilistically.

3See Fuchs (2007), Piskorski and Westerfield (2016), Smolin (2017), Zhu (2018) and Orlov (2019).

2

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associated with default. More generally, we show that disclosure opportunities have

heterogeneous real effects on financing and default, depending on whether they arise at

times when firms are setting up their capital structure or not. These effects also vary

predictably in the cross-section of firms and critically depend on market conditions.

Specifically, we build a model based on two ingredients. First, a dynamic agency con-

flict between firms and investors, due to the non-verifiable nature of cash flows. Specifi-

cally, managers can divert cash each period as in DeMarzo and Fishman (2007),4 which

implies that investors use capital structure and the threat of default as means to prevent

malfeasance by the firm’s insiders. Second, we introduce the option for firms to invest

in a costly technology that randomly generates information about future performance.

As in Dye (1985) or Shin (2003), once the technology is adopted, with some probability

the firm’s manager privately observes a signal that predicts future cash flows. While the

signal itself cannot be manipulated, it can be shrouded by the manager. The focus on

optimal contracting distinguishes our paper from the recent disclosure literature.5

The implementation proposed in DeMarzo and Fishman (2007) extends to our setting

with minor adjustments. In equilibrium the firm starts off by borrowing – both short and

long-term – and issuing equity. Dividends are payed after a sufficiently positive stream

of earnings realizations, while persistent bad performance brings the firm to default and

liquidation. Disclosure has the benefit of alleviating the moral hazard problem when

cash flows are low. On the one hand, the manager can rely on evidence to show that a

low performance is not due to diversion. On the other, the investors can condition their

liquidity provision on the disclosed evidence without distorting moral hazard incentives.

As a result, the funding liquidity of the firm at low cash flows depends on whether

or not disclosure occurs. The firm’s funding liquidity can be thought of as the unused

balance on its credit line, which evolves through time. In our model, the interest rate

on funds withdrawn from the credit line depends on both the reported cash flows, and

the evidence disclosed. Practically, such variations in rates can be thought of either as

covenants attached to the short-term debt issuance (e.g., Smith and Warner (1979)), or

as performance-sensitive debt (e.g., Manso, Strulovici and Tchistyi (2010)).

In particular, we find that – compared to the case of no evidence considered in De-

Marzo and Fishman (2007) – interest rates are zero when a low cash flow is preemptively

disclosed, while they are higher both when cash flows are low and there is no disclosure,

and when cash flows are high. In other words, the optimal capital structure features pay

4See also Bolton and Scharfstein (1990), Biais et al. (2007), and DeMarzo and Sannikov (2006).5See especially Acharya, DeMarzo and Kremer (2011), Guttman, Kremer and Skrzypacz (2014) and

DeMarzo, Kremer and Skrzypacz (2017).

3

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for verifiable bad luck. The result provides a novel rationale for ‘pay without performance’

(Bebchuk and Fried (2009)). Existing explanations tend to emphasize either capture of

boards by powerful executives, or the desire of motivating innovation by managers –

with the associated need to incentivize risk taking to compensate early failures (Manso

(2011)). In contrast, we show that it may be in the shareholders’ interest to compensate

the firm’s executives when bad performance is proven to be the consequence of bad luck.

This goal is more likely to be achievable with the advent of technologies that facilitate

the production and the analysis of performance-related quantitative evidence.

In light of the previous result, we can sign the effect of exercising the option on Pay-

for-Performance Sensitivity (PPS): it is negative.6 To see why PPS must fall when the

technology is adopted, consider first the PPS conditional on a signal being available. In

this case, disclosure provides insurance to the firm in bad states of the world, which

lowers the PPS. If, instead, the signal is not available, the PPS does not depend on

the technology, exactly as in DeMarzo and Fishman (2007). Because the actual PPS

is a convex combination of the two possibilities, it must decrease when the option is

exercised. In addition, we find that – unlike in previous models such as Biais et al. (2007)

or DeMarzo and Sannikov (2006) – in our model the PPS increases monotonically with

the firm’s past performance history: PPS rises as performance improves.

Conditional on a given performance history, firms that adopted the technology and

have a lower PPS are less likely to default, as one would expect. Indeed, credit spreads

are smaller when insiders are more likely to have evidence to disclose. Empirically, this

implies that – in secondary markets, when the firm’s capital structure is unchanged –

credit spreads are negatively associated with the frequency of voluntary disclosures. This

is consistent with the empirical findings of Balakrishnan et al. (2014), who consider

firms facing an exogenous drop in their information environment and react by enhancing

disclosure. Because major issuance events are unlikely to be frequent in their dataset,

they are likely to capture primarily the effect of disclosure on secondary markets, which,

consistent with our model, is positive.

Disclosure opportunities also affect the firm’s leverage and dividend payout. The firm

issues dividends only after a sufficiently positive sequence of shocks, which implies that its

dividend payout rates are positively correlated with its survival probability. Because firms

that disclose information more frequently have higher survival probabilities – conditional

on any given performance history – it follows that they also display higher dividend payout

rates. In contrast, because leverage is the highest when the firm is close to liquidation,

higher survival probabilities imply lower leverage. As a consequence, our model yields

6PPS is defined as the change in the insiders’ payoff over the change in cash flows across states.

4

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predictions about the effect of voluntary disclosure on the firm’s capital structure in

secondary markets.

Turning attention to primary markets – where capital structure choices are made

– a trade-off arises. On the one hand, disclosure alleviates the moral hazard problem

and increases a firm’s survival probability for any given level of liquidity. On the other,

disclosure lowers the value of providing the firm with liquidity in the first place, and so

it incentivizes investors to reduce the initial funding liquidity granted to the firm. As

a result, the firm may well end up on paths that entail lower survival probability when

the option is adopted. Which effect dominates depends critically on a firm’s profitability.

We are not aware of work testing this hypothesis directly, but the result could partially

account for the recent surge in average bond default rates (Becker and Ivashina (2019)).

In particular, we find that for high-profitability firms the beneficial effects dominate:

disclosure yields higher survival probabilities, greater funding liquidity and lower credit

spreads, as well as higher dividend payout rates and lower leverage. For low-profitability

firms, instead, the negative effects prevail: credit spreads widen, leverage increases and

dividend payouts fall. We solve analytically for a profitability threshold that separates

these firms, but the threshold is only sufficient (not necessary) for non-monotonicity

of credit spreads to arise. Firms more profitable than those at the threshold may be

negatively impacted by the option for some parameter configurations.

Empirically, our results suggest that studies of the real effects of disclosure should

pay attention to default risk, in addition to discount rates. While existing empirical

work uncovered a positive causal effect of voluntary disclosure on the liquidity of a firm’s

securities and on its cost of capital,7 our results suggest that such effects are mitigated

by the negative impact disclosure has on expected cash flows at low profitability firms,

which may be more likely to default when they are expected to disclose more frequently.

In contrast, the two effects are amplified at high profitability firms, where disclosure not

only lowers the cost of capital, but it also increases the expected cash flows generated by

the firm due to its reduced probability of default.

As for the patterns of adoption, we predict that the set of adopting firms consists

of those that experienced intermediate performance histories: the very profitable and

the very unprofitable ones do not adopt the technology. The region is characterized by

two performance-related thresholds. Below the lower threshold, the value of the firm as

a going concern is too low to justify spending resources on the technology. Above the

7See Francis, Nanda and Olsson (2008), Balakrishnan, Billings, Kelly and Ljungqvist (2014) andBoone and White (2015), and the surveys by Healy and Palepu (2001) and Leuz and Wysocki (2008).Bertomeu and Cheynel (2016) survey theory work that interprets such findings in CAPM-like models.

5

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upper threshold, the benefits of IT are too low, as the firm is already far from its default

boundary. In the adoption region, the beneficial effect of disclosure outweighs the cost

of exercising the option and investment is undertaken. As the cost of the technology

falls, the thresholds diverge and the adoption region expands. Importantly, this pattern

of adoption of our information-related real option is markedly different from that of

alternative physical options, such as those studied in DeMarzo, Fishman, He and Wang

(2012). Indeed, while in their model the value of the option is a monotonically decreasing

function of past firm performance, in our model such value is non-monotonic: it is the

highest for intermediate performance histories.

We also find that information technologies affect the valuation of some firms that did

not adopt them yet. More precisely, there exist two other thresholds – one below the

adoption region and one above it – such that firms located between these thresholds and

the adoption region strictly benefit from having the option to delay their adoption. Con-

cretely, low-performance firms wait to see their enterprise value increase, and condition

their adoption on receiving a positive cash flow shock, while high-performance firms keep

the option as an insurance policy, intending to adopt if they receive a sufficiently negative

cash flow shock.

The paper unfolds as follows. Section 2 reviews the related literature. Section 3

presents the economic environment and the contract space. Section 4 considers two finite

horizons versions of our model. A one-period example shows how evidence is irrelevant

for static incentives, suggesting that, if evidence plays a role, it must be that it affects the

dynamic incentive constraints. A two-period example shows how most of our results do

not rely on the horizon being infinite, and conveys the intuition for some of our findings.

Section 5 introduces the infinite horizon model. Section 6 discusses the impact of disclo-

sure on the policy dynamics and on other variables of interest. Section 8 implements our

optimal contract by means of short and long-term debt, and equity. Section 7 shows the

patterns of information technology adoption. Section 9 discusses the initiation problem,

when securities are issued. Section 10 concludes.

2 Literature Review

Our paper is related to several literatures. Theoretically, it builds on the dynamic agency

model developed by Clementi and Hopenhayn (2006), Biais et al. (2007) and DeMarzo

and Fishman (2007). A recent strand of papers on dynamic moral hazard introduced

information production and dissemination possibilities, and studied their consequences

on second best allocations (e.g., Fuchs (2007), Piskorski and Westerfield (2016), Smolin

6

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(2017), Zhu (2018) and Orlov (2019)). The distinguishing feature of our model is that,

while other papers focus on monitoring technologies where the principal acquires infor-

mation directly, we assume that the realized information is observed by the agent and,

to be payoff relevant, it must be voluntarily disclosed. This assumption captures the re-

alistic feature that most firms’ investors are institutional or retail, and do not participate

actively in the day-to-day operations of the firm.

Because we model information systems as technologies that produce disclosure op-

portunities for managers, a la Dye (1985) or Shin (2003), our work is also related to

the theoretical work on voluntary disclosure (e.g., Beyer and Guttman (2012), Acharya,

DeMarzo and Kremer (2011), Guttman, Kremer and Skrzypacz (2014), Marinovic and

Varas (2016) and DeMarzo, Kremer and Skrzypacz (2017)). While these recent papers

extended the Dye model to a dynamic setting, they differ from our setting in important

ways. First, managerial compensation is exogenous, whereas we consider optimal com-

pensation. This implies that their equilibria might feature strategic disclosure, whereas

ours do not, due to Homlstrom’s informativeness principle (Holmstrom (1979)). Sec-

ond, in some of these papers, evidence is potentially long-lived, and so the dimension of

analysis is not only what is being disclosed, but also when managers disclose.

Our paper also relates to a recent literature that focuses on understanding the het-

erogeneous effects of information technologies on the cross-section of firms. In particular,

Mihet and Philippon (2018) and Farboodi, Mihet, Philippon and Veldkamp (2019) fo-

cus on explaining the role of size in shaping adoption patterns and the consequences of

adoption for the price informativeness of stock listed firms. Our work is complementary

to this literature, in that we focus instead on how these technologies shape disclosure

patterns by managers and, ultimately, on how they impact the informational landscape

in which firms operate and interact with outside investors.

A related literature studies the consequences of real investment options for firms in

dynamic agency models (e.g., DeMarzo, Fishman, He and Wang (2012), Bolton, Chen

and Wang (2011)). Relative to this literature, we contribute by considering a different

type of option which, instead of directly increasing the firm’s cash flows, improves the

information available for the management to disclose the firm’s shocks to its investors.

As we discussed in the introduction, such options have very different effects.

Our paper is also related to the literature emphasizing the possible negative real effects

of a richer information environment. Most work on this topic assumes that the principal

receives some information, but cannot commit to how the information is going to be used

in determining some interim action (e.g., Cremer (1995), Meyer and Vickers (1997), Prat

(2005) and Zhu (2018)). In contrast, in our model the agent receives the information and

7

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needs to disclose it, while the principal has full commitment power.

Finally, at a more abstract level, our work is related to the literature discussing the

role played by hard evidence in mechanism design problems. Since the seminal work of

Bull and Watson (2004), the literature has recently flourished. Notable contributions are

Koessler and Perez-Richet (2014), Hart, Kremer and Perry (2017) and Ben-Porath, Dekel

and Lipman (2019). In this context, we are the first – to our knowledge – to consider the

role of evidence in an otherwise standard dynamic agency setting.

3 Environment

A firm produces i.i.d. cash flows xt ∈ {h, l} for t = 1, 2, .., T , where h > l > 0. Define

∆ := h − l, p := P (xt = h) ∈ (0, 1), and µ := E(xt). The firm is owned by a Principal

(P) – who represents the firm’s investors – and is operated by a Manager (M). Both P

and M are risk-neutral and discount future consumption at the same rate r ∈ (0, 1).8

Moral hazard. We introduce the possibility of moral hazard by assuming that M

privately observes the realized cash flows {xt}. By misreporting a good cash flow, claiming

it to be bad, M can divert ∆ output and obtains a private benefit of δ := λ∆, where

λ ∈ (0, 1] represents the severity of the moral hazard problem.9 From the revelation

principle, we can restrict communication protocols to direct messages that report xt, and

focus on the implementation of truthful reporting.

Evidence. We assume that P can choose to invest in an information technology

that produces evidence et ∈ {g, b} each period with probability π ∈ (0, 1). To ease

notation, π in the paper denotes a random variable that takes values of either 0 or π,

depending on whether the technology has been adopted (π = π), or not (π = 0).10

To make this investment, P must spend a fixed cost of c ≥ 0.11 Evidence consists of

verifiable information that cannot be manipulated, and which perfectly predicts cash

flow xt: good evidence implies high cash flows, while bad evidence implies low cash flows.

IT adoption effectively corresponds to the exercise of a one-time American option with

infinite maturity that cannot be reversed, with strike price c.

Once the option is exercised, P expects evidence to be available with probability

8Common discounting is not needed to derive our qualitative results, but it simplifies the arguments.9The notation here is not redundant: the effects of ∆ on allocations and contracts are slightly different

from those of λ, in ways that we will emphasize while discussing the comparative statics.10The fact that in the absence of technological investment π = 0 is just a normalization. All our

qualitative results go through unchanged if we assumed that, absent investment, the firm would have apositive π < π.

11The cost can be thought of as the presented discounted value of the setup and maintenance expenses.

8

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π, but she never knows whether M possesses evidence or not. So, at each date t, M

chooses whether or not to voluntarily disclose the realized evidence to P. We denote

the disclosure action by at ∈ A := {d, n}, where d stands for disclosure and n for non-

disclosure. If M discloses the evidence, investors will predict the cash flow accurately.

That is, p(xt = h|et = g) = p(xt = l|et = b) = 1.12 Because (i) disclosure is always

incentivized, and (ii) the availability of evidence is conditionally independent from the

realized cash flow, non-disclosure has no impact on the investors’ beliefs. That is, absent

evidence disclosure, P predicts that cash flows are high with probability p.

Contracting. To maximize investors’ value, P offers M a contract that specifies, for

every history of reports and disclosures, the probability of liquidating the firm θt ∈ [0, 1],

and the cash compensation ut ≥ 0. In the first best case, the firm is never liquidated

and has the value of s∗ := µ(1+r)r

. If the firm is liquidated, both parties get their outside

option payoff, which is normalized to zero. Figure 1 shows the timing of events in a

generic period t, prior to the exercise of the IT-investment option.

Figure 1: Timing in period t, prior to exercising the option

t

P exercises or delays

the option to invest

Liquidation

with prob. θt

M voluntarily discloses M reports cash flow M is paid ut

t+ 1

4 Finite-horizon model

To highlight the key driving forces behind our results, we start with a static and a two-

period versions of the model. For simplicity, we set r = 0 in this section.

12Since the effects of evidence on our outcomes of interest are already non-monotone and complex withperfect evidence, it does not seem necessary to also consider imperfect correlation in this model.

9

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Figure 2: Event tree of static setting

d

n

d

h

h

l

l

udh

unh

disclosure incentive

unl

udl

disclosure incentive

diversion incentive

One-period setting. Figure 2 draws the event tree when T = 1. The set of possible

outcomes is H1 := {dh, dl, nh, nl}, and cash compensations to M are denoted by ui, for

i ∈ H1. The contract must provide two kinds of incentives: (i) to prevent the agent from

diverting cash flows, which requires unh ≥ δ + unl; (ii) to disclose information, which

requires both udh ≥ unh and udl ≥ unl. It is optimal for P to set unl = udl = 0 and

udh = unh = δ, and since c > 0 the option is never exercised.

Two-period setting. It follows from the one-period case that at t = 2 evidence is

irrelevant. So, the set of relevant final histories is H2 := {ahh, ahl, alh, all}a∈A, where

the first element a ∈ {d, n} denotes M’s disclosure action in the first period; the second

and third elements denote the t = 1 and t = 2 realized cash flows, respectively. As in

most dynamic agency models (e.g., Biais et al. (2007)), committing to liquidate the firm

when x1 = l may be optimal, because it alleviates the diversion problem and reduces

the rents required for incentive compatibility to hold. Now, disclosures matter and the

optimal contract is shaped by both the cost c and the benefit π of exercising the option.

Proposition 1. If T = 2, there exists a c such that if c ≥ c the option is never exercised,

while if c < c there exist two profitability thresholds p and p such that p < p and:

(a) If p ∈ [p, p], the option is exercised and the probability of default is (1− p)(1− π);

(b) If p < p, the option is not exercised and the firm’s probability of default is zero;

(c) If p > p, the option is not exercised and the firm’s probability of default is 1− p.

Because ∂p/∂c > 0 and ∂p/∂c < 0, a reduction in the strike price of the option c increases

the probability of default of low profitability firms, while it increases the probability of

default of high profitability firms.

10

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In the two-period case, evidence might enable P to distinguish bad luck from bad

behavior in the first period, and affect the optimal termination policy. Figure 3 shows

that the option to invest at a strike price c and produce evidence with probability π

attracts firms that are neither too profitable (p ≤ p), nor too unprofitable (p ≥ p).

When the two value functions (conditional on whether the option is exercised or not) are

tangent, the option is only exercised by firms with p = p. If the cost drops to c < c, the

set of firms that exercise the option expands to p ∈ [p, p]. High profitability firms choose

not to invest and terminate when x1 = l. Low profitability firms, in contrast, never

terminate and do not exercise the option either. If c > c the option is never exercised.

Figure 3: The set of firms that exercise the option as the strike price c changes

ppp=0 p=1p

ValueFunctions

c<c c=c c>c No Option

Thus, a reduction in the strike price of the option c leads to increased adoption and

more disclosure by both profitable and unprofitable firms. However, Figure 4 shows

that its effect on default probabilities and credit spreads is heterogeneous across firms.

For high-profitability firms that switch to exercising the option (right-panel), default

probabilities decrease as disclosure avoids inefficient termination. For low-profitability

firms that switch (left-panel), the opposite occurs. While at a high cost c they had a

low (zero) default probability, as c drops evidence provides a tool for P to reduce the

rents paid to M, while not defaulting the firms when disclosure occurs. As a result, the

firm’s default probability rises. Together with the cost c, this amounts to an increase in

the deadweight losses and a reduction in the social surplus, even though it increases the

investor’s payoff.

11

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Figure 4: Private and social returns from IT adoption

The two-period model is stylized, in that either the firm exercises the option imme-

diately, or it never does. The timing does not allow one to study how the possibility of

delaying the exercise time shapes the dynamics and the initial conditions. However, it

highlights the complex interplay between the strike price of the option, the frequency of

disclosure and the firm’s default risk. In the next section, we analyze these forces in the

full infinite-horizon model, where the option to adopt the information technology can be

exercised with delay.

5 Infinite-horizon model

In this section, we first formulate the firm’s problem in the infinite-horizon environment,

and then characterize policies and their dynamic features.13

5.1 Contracting

As is well known, when shocks are i.i.d., the agent’s continuation utility v is a state

variable that summarizes all relevant information in any given history. For any state v,

the contract specifies the probability of liquidating the firm at the beginning of the period

θ, and then compensates M either with cash, or with promised utility contingent on M’s

actions. When evidence is disclosed, the contract pays ud = (udh, udl) ∈ R2 to M and

promises continuation utility wd = (wdh, wdl) ∈ R2, depending on whether the high or the

low cash flow is reported. Similarly, when no evidence is disclosed, the contract pays M

cash either un = (unh, unl) ∈ R2, and promises continuation utility wn = (wnh, wnl) ∈ R2.

13More rigorous arguments which guarantee that the recursive representation of our problem is appro-priate are standard and so we leave them to the Appendix.

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Whether it is worthwhile to invest in the costly information technology or not, and if

so, when to make the investment, all depend on the value that this option brings to the

firm. To evaluate the moneyness of this evidence-generating option, we first consider the

optimal contracting for the firm given the investment has already been made. We then

step back and determine the optimal option exercise patterns.

Given that the investment in the information technology has been made, evidence

regarding future cash flows arrives with probability π. Because our programming that

solves the firm’s policies in this scenario also applies to the scenario where evidence never

arises (or the investment option is never exercised), we use the variable π to represent

both, with the indication of π = π for the former and π = 0 for the latter.

Before we define the firm’s problem, we consider the diversion and disclosure incentive

constraints. First, since the manager can always conceal evidence, any voluntary disclo-

sure has to be contractually incentivized. Contracts may disregard evidence in some

states of the world. However, because of Holmstrom’s informativeness principle, it only

makes sense that evidence disclosure is either promoted, or overlooked; it should never

be actively prevented. That is, whenever the manager obtains good evidence:

udh +wdh

1 + r≥ unh +

wnh1 + r

(ICg)

Likewise, whenever the manager obtains bad evidence we have:

udl +wdl

1 + r≥ unl +

wnl1 + r

(ICb)

Second, when the manager does not disclose good evidence, he can always report a low

cash flow and divert ∆. So, the diversion incentive compatibility demandszAA:

unh +wnh

1 + r≥ δ + unl +

wnl1 + r

(ICn)

Any feasible contract must fulfill its promises and deliver the given continuation value.

In other words, the optimal contract satisfies a promise-keeping constaint which requires:

v = (1− θ)[πEd

(ud +

wd

1 + r

)+ (1− π)En

(un +

wn

1 + r

)], (PK)

where, to ease notation, we define M’s expected utility conditional on evidence disclosure

as Ea(ua + wa

1+r

)= p(uah + wah

1+r) + (1− p)(ual + wal

1+r) for a = d, n. In addition, contracts

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must satisfy limited liability, i.e.:

udh, unh, udl, unl ≥ 0 (LL)

Because the agents share the same discount factor, it follows that the optimal contract

from P’s perspective also maximizes firm value (i.e., surplus), given a utility v promised

to M.14 Thus, the optimal contract solves the following dynamic program:

s(v) = maxθ,uj ,wj

(1− θ){µ+

1

1 + r

[πEd(sd) + (1− π)En(sn)

]}(S)

s.t. (PK), (ICg), (ICb), (ICn), (LL),

where s(v) denotes the expected firm value, sa = (s(wah), s(wal)) for a = d, n, and

Ea(sa)

= ps(wah)+(1−p)s(wal) denotes the expected firm values conditional on possible

disclosure actions.

The objective function of (S) reflects the fact that (i) with probability θ, liquidation

takes place before the subsequent evidence and cash flow realize, in which case the firm

value drops to zero; and (ii) with probability (1− θ) the firm is not liquidated, in which

case the firm value depends on whether M receives the leading evidence or not, and

whether the cash flow is high or low. Because the two events are independent, we can

express the expected firm value as that in the objective of (S).

On the one hand, the program (S) with π = π solves the firm’s problem given the

investment option has already been exercised. On the other hand, if π = 0, the program

exactly solves the case where the option is never exercised (no evidence ever possible).

This is because in the latter case, the only relevant control variables are those conditional

on no disclosure. Hence, we use s(v; π) and s(v; 0) to denote the value functions of (S)

for these respective cases.

5.2 Investment option

We next analyze the investment decision, i.e. the decision of whether and when to exercise

the investment option. Suppose that, for a given history represented by v, the firm has

not yet exercised the option. The firm’s value in this scenario is denoted as f(v) and,

obviously, no evidence will be disclosed today. If the firm is not liquidated – which occurs

with probability (1 − θ) – then it obtains the expected cash flow µ today and proceeds

14This does not imply that a contract that maximizes P’s expected utility is socially optimal: ingeneral, P starts the contract from a socially suboptimal initial condition – we shall return to this point.

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to tomorrow’s state of either wnh or wnl, depending on the cash flow reported by M.

Come tomorrow, the firm can either invest c and obtain the value of s(wni)−c (where

i = h, l) from the subsequent date onwards, or delay investment again and obtain the

value of f(wni). It is easy to see that when s(wni) − c > f(wni), the firm exercises

the investment option tomorrow. Otherwise, it waits until at least one more period to

invest. Hence, the firm’s problem when the investment has not been undertaken yet can

be formulated as follows:

f(v) = maxθ,uj ,wj

(1− θ){µ+

1

1 + rEn[

max(fn, sn(π)− c)]}

(F )

s.t. (PK), (ICn), (LL)

where fn = (f(wnh), f(wnl)), and π = 0 in (PK).

5.3 Initiation and payout

When the firm is initiated at time zero, P promises a continuation utility v0 to maximize

its expected profits over the lifetime of the firm. That is,

v0 = arg maxv{max[f(v), s(v; π)− c]− v} (1)

Clearly, at the outset, the firm may either exercise the option right away or wait to make

the investment later.

Because liquidation is inefficient, it may be optimal to delay M’s cash compensation

until the continuation utility v is sufficiently large. Throughout the paper, we adopt the

payout policy that M is paid by cash, if the firm is indifferent between paying him or

delaying the payment, which is without of generality. Formally, we define the cash payout

boundary as the smallest continuation utility where the firm value reaches its first best.

That is,

v := inf{v : f(v) = s∗ or s(v; π) = s∗} (2)

The definition implies that the firm value (with or without evidence) is strictly less

than the first best s∗ before the continuation utility reaches v. In general, both the payout

boundary and M’s payoff dynamics may depend on the availability of evidence. The next

result shows that actually the value v is a constant, irrespective of evidence availability,

but the cash compensation varies with the option exercise strategy and level of π in the

short-run.

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Proposition 2. The cash payout boundary is:

v = r−1(1 + r)pδ. (3)

Moreover, for a ∈ {d, n}, the optimal cash compensation is

ual(v) = 0, uah(v) = max{

(1 + r)δ − (1 + r)(v − v), 0}

(4)

where r(π) := r1−(1−p)π .

Proposition 2 shows that the payout boundary v does not depend on the possibility

of generating and disclosing evidence. Even when the cost of generating evidence is

infinitely large (or π is infinitely small or zero), the boundary does not change. Then, no

cash payment is made to M if low cash flow is reported. When the firm is one-step away

from v, M receives cash compensation upon reporting high cash flow. In addition, the

result says that the cash payment is reduced if evidence is more likely to be available.

However, when the firm reaches v, the cash compensation is independent of evidence.

The case of π = 0 in (4) indicates that the investment option has not been exercised.

The investment option has no impact here, because after this cash payment, the firm is

at v where the investment will not be made.

This is intuitive: the payoff boundary v is the smallest continuation utility at which

incentive constraints cease to bind. Once that boundary has been reached, evidence is

no longer useful. Because at v the firm is never liquidated, cash compensation is not

delayed further. As long as evidence is uncertain, the magnitude of cash compensation is

δ, independent of evidence, whenever good performance reported, because M can always

disguise good news and divert cash flows.

6 Impact of evidence disclosure

The decision to invest in the information technology depends on the value added by the

availability of evidence IT brings about, net of the strike price c. In this section we first

characterize the firm’s problem given that the investment has already been made. In

the next section we examine which firms exercise the option and under what conditions

they exercise. To highlight the role of evidence disclosure, we consider what happens

if the evidence is more or less available (the intensive margin), and then contrast the

policies with the benchmark case as in DeMarzo and Fishman (2007) where evidence is

never available (the extensive margin). This benchmark case corresponds to our model

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in which the option is never exercised.

6.1 Policy characterization

We first characterize the firm’s problem (S). Recall that it solves two possible scenarios:

the option already exercised (π = π), and the option never exercised (π = 0).

Before reaching the payout boundary, M is incentivized by variations in her promised

continuation values. If the continuation value ever grows high enough, the firm is never

liquidated, all constraints of the firm’s problem become slack, and firm value reaches the

first best. When M’s continuation value is at intermediate levels, liquidation may occur

after a sequence of low cash flows and no disclosures. When the continuation value is low

enough, the only way to both align incentives and fulfill commitment is to stochastically

liquidate the firm at the beginning of the period. To characterize the dynamics, we define

the thresholds such that no liquidation can possibly occur in the next n periods to be:

vn := inf{v : no liquidation in at leastn periods}, for n = 0, 1, 2...

These values correspond to the lowest continuation values such that the firm can

survive with certainty for at least n periods. For example, if v > v1 the firm will not

be liquidated in the current (or one) period, but may be liquidated in the next period.

These thresholds are related to the previous definitions of liquidation probability θ and

the payout boundary v. Specifically, stochastic liquidation at the beginning of any period

is positive (θ(v) > 0) if and only if v < v1. In addition, the payout boundary v is the

limit of this sequence of thresholds v∞; indeed, liquidation ceases if v ≥ v.

Because the firm can be liquidated, any randomization of continuation utility is costly

for both parties, implying that the firm value s(·) is concave. Using concavity and the

optimal conditions of the firm’s problem (S), we can show which constraints bind and

derive the optimal policies.

Lemma 1. For any v < v in the firm’s problem (S), the constraints (ICg) and (ICn)

bind while (ICb) holds as strict inequality.

One can immediately see that, contingent on the high cash flow being reported, ev-

idence is payoff irrelevant: wdh = wnh. In other words, as long as the investors receive

a high cash flow, the payoffs to both M and P are not affected by evidence disclosure.

Contingent on a good performance being reported, M does not divert and hence there is

no need to further condition payoffs on evidence disclosure. Therefore in the rest of the

paper we do not distinguish M’s payoff across states dh and nh. Accordingly, we denote

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wh and uh as the continuation value and the cash payment, respectively, conditional on

cash flows being high. Notice that this property would not hold in a monitoring setting,

in which P observes the realized signal directly. In that case, P could further reduce the

payment to M when evidence is available, without violating any incentive constraint.

In contrast, the optimal contract provides strict incentives for M to disclose bad news:

i.e., wdl > wnl. Punishing M for a bad performance is costly to P because it induces more

inefficient liquidation. If the evidence shows that the bad performance is not caused

by M’s behavior, but instead by bad luck, then M should not be punished. Promising

M higher utility in the state dl does not worsen the diversion problem, but improves

efficiency by reducing the probability of liquidation.

Given the active constraints and the optimality conditions of the firm’s problem (S),

we obtain an explicit solution for the optimal policies:

Proposition 3. The optimal policies for the firm are as follows:

• For v ∈ (0, v1]: θ = v1−vv1

, wnl = 0, wdl = v1, wh = min{rvp, v}

;

• For v ∈ (v1, v]: θ = 0, wnl = v − r(v − v), wdl = v, wh = min{wnl + rv

p, v}

;

• The n-period liquidation thresholds are vn(π) = [1− ( 11+r

)n]v.

If stochastic liquidation does not occur at the beginning of the period, the policies

wi(v) for i ∈ H1 are the same as wi(v1). In addition, firm value in this region is linear.

Given this characterization, we clearly see that – after a low performance – the contract

possibly promises the manager a higher continuation utility when bad news is disclosed

(wdl = v1 > v), deviating significantly from the case of π = 0 considered in previous work.

In this region, whenever M discloses evidence of transitory bad luck P compensates for

disclosure by raising the promised utility to v1, independently from the continuation

utility entering the period. The reward depends on v1 − v.

In the region above v1, M is still rewarded for disclosing bad luck: wdl = v.15 The

contract forgives the low performance today, and starts tomorrow as if the history is

the same as before the current low cash flow. This mechanism does not affect the M’s

diversion incentives, because M can never mimic the type who discloses evidence that the

cash flow will be low. Moreover, volatility in continuation utility is costly for investors

because liquidation is inefficient. So, it is optimal to set wdl as close to v as possible.

Finally, as standard, the optimal contract rewards good luck. Proposition 3 shows

that the ranking of continuation utility does not depend on the levels of v and π. M gets

15Notice that to keep the continuation utility fixed across times effectively requires a payment from Pto M, because of the time value of money.

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the largest continuation utility contingent on high performance, the lowest one contingent

on low performance and no disclosure, and the middle one contingent on disclosure of bad

news. This pattern implies that, on the fastest route to liquidation, M never discloses

evidence and always reports low cash-flow. So, the n-period threshold can be explicitly

derived from the policy functions. Evidently, both the liquidation thresholds and the

policy dynamics depend on M’s disclosure behavior and on the availability of evidence.

6.2 Comparative statics

Having characterized the optimal policy functions, we can examine how the optimal

contract varies with the probability that the information technology produces evidence.

These comparative statics highlight the impacts of the quality of the information technol-

ogy. We focus on the impact of evidence disclosure on firm value, default risk, managerial

compensation and firm dynamics. This analysis is specific to dynamic models because,

regardless of the initial conditions of the problem, any v ∈ [0, v] is on-the-equilibrium

path. That is, there always exists a sequence of shocks that can take the firm from v0

to any such v. Before presenting the results, it is useful to provide a formal definition of

both credit spreads and Pay-Performance Sensitivity (PPS). We follow the literature and

define PPS as:

PPS :=E(v |x1 = h)− E(v |x1 = l)

∆(5)

This measure indicates in percentage terms how M’s compensation changes with firm

performance. Normally, we have that Credit spread = (1− recovery rate)× Pr.[default],

but because we normalized the recovery rate to zero,

Credit (or CDS) spread = 1− s

s∗, (6)

where s∗ denotes the first best value of operating the firm, and s denotes the value at the

constrained best, as implemented by our optimal contract (i.e., the sum of the principal’s

and the manager’s expected payoff). Importantly, this is distinct from the agency cost,

which would be defined as Credit spread + managerial rents.

Proposition 4. When the availability of evidence π is higher, the optimal contract ex-

hibits the following comparative statics, for any given v < v:

(a) firm value s increases or, equivalently, its credit spread falls;

(b) pay-performance sensitivity falls, while it increases with v for any given π

(c) wh and wnl both weakly fall, while wdl stay constant;

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(d) the n−step liquidation threshold vn falls, for n = 1, 2, ...

0 0.5 1 1.5 2

v

0

1

2

3

s(v)

Firm Value

=0

=0.5

=0.9

0 0.5 1 1.5 2

v

0

1

2

s(v)

- v

Investor Value

=0

=0.5

=0.9

Figure 5: Firm and Investor Value

Part (a) of Proposition 4 shows the overall effect of evidence on firm value and investor

payoff. Given a promised utility to the manager, more evidence disclosure increases

expected firm value s(v). It follows that the investors’ expected payoff, s(v) − v, rises

with π, for any given value of v. Figure 5 plots numerical examples of these effects. The

result also clarifies that the investors’ time-zero payoff (i.e., the highest value in each

curve of the right plot) must increase with the availability of evidence. This is because

investors choose where to start the firm so as to maximize their time-zero payoff.

0 5 10 15 20

V

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Liq

uid

ation P

robabili

ty

π=0

π=0.5

π=0.9

Figure 6: Simulated Liquidation Probability

To clarify how evidence affects the probability of default in secondary markets, i.e.,

conditional on a given performance history v, we simulate the policy dynamics as shown

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in Figure 6. The figure illustrates a cross section of firms that are identical except for

having different chances of evidence disclosure. After certain histories, if these firms

promise the same expected utility to their managers, then a firm with a higher chance

of disclosing evidence has a lower probability of being eventually liquidated. This is the

main reason why social efficiency and investors’ payoff both improve.

We can explicitly characterize the firm’s PPS from Proposition 3 as:

PPS = λ− πr [v −max(v, v1)]

(1 + r)∆(7)

Obviously, this measure depends on both v and π. An example is displayed in Figure

7. When π = 0, the PPS measure equals λ, regardless of v. This result holds in existing

models of dynamic moral hazard without evidence disclosure – e.g., DeMarzo and Fish-

man (2007). Once we introduce evidence, a few effects emerge that are worth stressing.

First, the PPS decreases with π, suggesting that cash flow incentives and disclosure are

substitutes in the optimal contract. More evidence reduces PPS because the manager’s

payoff is unassociated with low performance if the evidence predicts its bad luck. Second,

the PPS also depends on v itself. In particular it increases with v, converging to λ as v

approaches v. Larger PPS induces higher variations in continuation value, exacerbating

the inefficient liquidation. When v is small, liquidation is more severe, making a large

PPS more costly for the firm. Allowing evidence disclosure significantly reduces PPS and

the chance of liquidation. Evidence disclosure has no effect on PPS only at the payout

boundary v, where the firm has no chance of being liquidated, .

Previous results give the explicit forms of the policies {wi}i∈H1 , θ, and the n-period

liquidation boundary vn. Part (c) of Proposition 4 analytically illustrates the properties

of the firm evolution when the disclosure frequency π varies.

v v

PPS(v|π)

π = 0π = 0.5π = 0.9

Figure 7: Pay-for-Performance Sensitivity

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The result shows the impact of disclosure on how continuation utilities evolve. The

managerial payoff wdl either remains the same as the beginning of period utility v, or it

jumps up to v1, regardless of the level of π. From the promise-keeping constraint, we

know that both wh and wnl must fall (weakly if v < v1), because the continuation utility

is more likely to stay at v. The diversion incentive constraint binds, establishing that the

gap wh − wnl must be constant and equal to (1 + r)δ. Figure 8 illustrates this pattern

of how policies move as π increases for a given value of v (or a given history). As π

increases, the continuation utility is less likely to move downward, but the lowest value

becomes worse.

(1−p)(1−

π)

v

wh

wdl

wnl

(1−p)(1−

π ′)

w′h

w′dl

w′nl

π π′ > π

x

x

vv

p p

(1− p)π (1− p)π′

Figure 8: Impact of Evidence on Managerial Payoff

Finally, part (d) of Proposition 4 shows that the availability of evidence has a down-

side: the n-period liquidation thresholds vn all increase as evidence become more avail-

able. For the cross-section of firms starting at the same continuation utility, the shortest

time to be liquidated becomes shorter as π goes up. In other words, on the fastest way

to liquidation, a firm with a better evidence technology can be liquidated faster. In con-

trast, the shortest time to reach the payout boundary becomes longer as π goes up. These

patterns are plotted in Figure 9, which verifies the previous illustration that the lowest

continuation utility becomes lower as π increases. According to the characterization in

Proposition 3, we know that the fastest way to liquidation on the equilibrium path occurs

when a firm never discloses evidence and experiences a sequence of low cash flows.

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0 5 10 15 20

V

0

5

10

15

20

25

30

35

Tim

eShortest Time to Liquidation

π=0

π=0.5

π=0.9

0 5 10 15 20 25 30 35 40

V

0

1

2

3

4

5

6

Tim

e

Shortest Time to Payout ( )v

π=0

π=0.5

π=0.9

Figure 9: Simulated convergence time toward the two boundaries

7 Investment option

The value of evidence in our model comes from two channels. First, as described before,

the availability of evidence increases firm value by avoiding inefficient punishment on

M. Second, because the value of evidence is endogenous and varies with continuation

utility, the optional delay of paying the cost can be valuable to the firm. Clearly, if the

investment cost is too large, then the option is never exercised. The largest cost at which

the investment option is ever exercised is given by:

c = maxv{s(v; π)− s(v; 0)} (8)

Moreover, the option is not exercised right away either in the region close to v or in the

region close to 0. Accordingly, there are two thresholds that reflect these two regions.

vl = infv{f(v) ≤ s(v; π)− c}, and vh = sup

v{f(v) ≤ s(v; π)− c} (9)

Proposition 5. The threshold cost is c > 0. Both c and vh increase in the evidence

availability π, but vl decreases in π. Further, vh decreases in the investment cost c, while

vl increases in c.

If the firm never exercises the investment option, its value is s(v; 0) given a history

represented by v. Obviously, the firm value with the investment option f(v), as defined

in (F ), is no less than its baseline value of s(v; 0). Therefore, the value of exercising the

option s(v; π) − f(v) is smaller than s(v; π) − s(v; 0), which is further smaller than the

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cost if c > c. Hence, the option is never exercised. The value c reflects π and other

parameters including the severity of agency λ, the profitability of the firm p, and so on.

When c < c, the investment option is possibly exercised at certain v. When the

continuation utility v is close to v, the probability of default becomes very small. Evidence

that prevents the inefficient termination becomes not that valuable. Hence, the value of

delaying investment f(v) becomes very close to the first best level s∗, which is strictly

larger than the value of exercising the option right away s(v; π) − c. When v is close to

0, the firm value is small with or without evidence disclosure. The value of exercising the

option right away s(v; π) − c becomes tiny or even negative, which is smaller than the

value of waiting to invest. The following result summaries the possibility of exercising

the investment option, and when to exercise it.

Proposition 6. The firm never exercises the investment option if c ≥ c. Otherwise,

there exists a nonempty interval v ⊂ (vl, vh) where investment option is exercised right

away, while the option is delayed for v ∈ [0, vl] ∪ [vh, v].

0 2-0.5

0

2

s∗

vhv

l v

Never exercise

Delay exercise

Exercise now

Exercise

Delay

Delay

Figure 10: Option Exercise Region

Figure 10 illustrates the result of Proposition 6. It plots a numerical example of

different firm values over continuation utility when c < c. The green line plots the firm

value if the option is never exercised which is s(v; 0) or the case of DeMarzo and Fishman

(2007). The blue line plots the firm value if the investment option is exercised right

away which is s(v; π) − c. The red line plots the firm value f(v) of delaying the IT

investment. The investment is made if v ∈ (vl, vh), but delayed if v ≥ vh or v ≤ vl. The

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difference between the red and green line in Figure 10 reflects the option value of the IT

investment. This difference is large at intermediate levels of v, because for these levels

of continuation utility, the firm is likely to spend more time in the constrained stage and

therefore more likely to exercise the investment option eventually. On the left and right

tails, this difference shrinks, because the firm is likely to be liquidated or to reach the

first best, not exercising the investment option. When this option value becomes large

and surpasses the value of investing right away, it is better to delay and wait.

Proposition 5 shows that if evidence is more available, then the firm is more likely to

exercise the investment option overall, and the region of delay to invest shrinks. If instead

the investment cost is higher, then the firm is more cautious, or more likely to delay the

investment until its accumulative performance moves to a smaller middle range.

8 Capital Structure Implementation

This section implements the optimal contract using standard financial securities. To

facilitate comparison with dynamic moral hazard models that do not have the possibility

of disclosure (e.g., DeMarzo and Fishman (2007)) the securities in our implementation

only include equity, long-term debt, and a credit line (short-term debt).

The long-term debt claim is essentially a perpetuity that pays a fixed coupon every

period forever. The credit line defines the amount of credit that can be withdrawn by

the firm anytime within the (endogenous) limit z. The firm’s debt capacity, which is

the difference between the credit limit and its balance, proxies the firm’s liquidity level.

Finally, the equity component is a claim against the firm’s dividend payments. Given

any capital structure, the manager controls the firm’s liquidity and payout policies. More

precisely, the manager determines how and when to withdraw from (or repay to) the

credit line, and how and when to pay dividends.

Within this set of securities, disclosure affects the evolution of the credit line and. In

particular, it determines its interest rate. In our model, any balance on the credit line

account is charged an interest rate ri, for i ∈ H1, that is contingent on both performance

and disclosure. In contrast to models such as DeMarzo and Fishman (2007), here investors

may sometimes charge a higher interest rate than r, or they can forgive part or all of the

current period interest charge.

The credit account balance reflects any repayment at the beginning of each period

before the firm cash flow realizes. The following result summarizes a security design that

implement the optimal contract.16

16Evidently, as in all other security design problems, such design can never be unique.

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Proposition 7. Under the following security and compensation design, the manager

always discloses any evidence that might be available, and cash flows are used to either

repay coupon and credit balance or to issue dividends.

• The manager holds λ fraction of the firm equity;

• the long-term debt coupon is l;

• the credit line has limit z = vλ

, and contingent interest rate rdl = 0 and ri 6=dl = r(π).

The firm only issues dividends after it pays off the credit balance and the coupon.

In the implementation, the credit balance or borrowed short-term debt, denoted as

m, summarizes the history and functions as the state variable. It maps one-to-one to the

state variable v of the firm’s problem (S). On the one hand, the manager can borrow

all the available credit and pay it out as dividend. Thus, the continuation value of the

manager in the firm must be at least λ(z − m). On the other hand, the investors will

not leave more information rent (in the form of liquidity) than necessary to the manager.

Hence the continuation utility of the manager must be

v = λ(z −m) (10)

which must hold at any history. Given this relation, as well as the policy dynamics in

Proposition 3, we can figure out how the firm’s short-term debt evolves, which further

implies the interest rates specified in Proposition 7.

To further illustrate the mechanisms, let us consider how the credit balance evolve

over time. Suppose that the firm starts certain period with credit balance m. It pays

the long-term debt coupon l from the cash flow. The interest rate on the credit line is a

constant value r unless bad cash flow news is disclosed when the interest rate becomes

zero. The credit balance in the following period denoted as mi∈H1 will be

mh = (1 + r)m+ (1 + r)(dh −∆) (11)

mnl = (1 + r)m (12)

mdl = m (13)

where dh = uhλ

is the dividend payout. If a bad news is disclosed, then interest is forgiven

in the current period, and the new balance will stay the same. If the high cash flow

realizes, the firm is charged a interest rate of r but will have (1 + r)∆ more cash to repay

the short-term debt in the next period (independently from disclosure). Therefore, the

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new credit balance mh follows (11). If low cash flow realizes and no evidence disclosed,

then r is charged toward the beginning balance m and the new balance mnl follows (12).

As shown in Proposition 7, one important feature of our model is that the equity

holdings, the long-term debt coupon, and the credit limit do not depend on the availability

of evidence: only the short-term interest rate does. The equity holdings determine how

the residual cash flow (or dividends) are split between the manager and investors. In our

model, when the firm starts paying out dividends, it has no possibility of being liquidated

and surplus reaches the first best. In that stage, evidence is disclosure is payoff irrelevant.

The necessary way to incentivize the manager is for him to hold λ fraction of dividend

payments.

However, the interest rates of the credit line affect the evolution of the firm’s short-

term debt holding. Since evidence disclosure does affect firm liquidity in the short-run

transition, the interest rates must vary with the manager’s disclosure decisions. The

variation in interest rates is essentially to incentivize the manager to disclose bad news.

It is easy to see from Proposition 7 that the average interest rate is exactly r, but the

interest gap between disclosing bad news or not is rdl− rnl = r, and it increases with π. In

other words, as the probability of the manager possessing evidence increases, investors on

average still earn the risk free rate r, but they will design a larger interest rate variation

to induce disclosure of bad news.

9 At issuance date

Now that we have characterized the optimal contract, we can examine the agent’s rent, the

firm value, and the default probability at the issuance date (or time zero). In particular,

we will show how these values change as the cost of producing evidence varies.

Proposition 8. If p is smaller than some p ∈ (0, 1), then at the issuance date, the agent’s

rent v0, the firm value max{f(v0), s(v0; π)−c}, and the credit spread can be all increasing

or all decreasing, in the investment cost c.

In general, the time-zero properties are hard to characterize, because they reflect

the expectation of all future firm performances and evidence disclosures. Note that the

firm’s optimal starting point v0 depends on the marginal value of raising continuation

utility to the firm value. The key feature of our model is that as evidence becomes more

available this marginal value (at v0) can be either larger or smaller. On the one hand, the

continuation utility is less likely to move downward as π increases, making the marginal

value of continuation utility smaller. In this case, the firm initiates by granting the agent

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lower expected rent, which may drive the initial firm value downward. On the other

hand, if there is no disclosure, the continuation utility can drop to a lower level, leading

to a smaller firm value. The marginal value of continuation utility can become larger to

hedge such situation. In this case, the firm initiates at larger v0, which implies a larger

firm value at issuance.

It’s not hard to understand that as generating evidence becomes cheaper the firm value

can be higher because P has an additional channel to govern the agency conflict. But

the opposite can also be possible. In fact, Proposition 8 confirms the intuition obtained

in the two-period model that the credit spread of low profitability firms may actually

increase as disclosure becomes possible. Managers who are expected to have access to

evidence may be worse off than the less informed ones.

0 0.1 0.2 0.3 0.4 0.5

c

0.08

0.09

0.1

0.11

0.12

0.13

0.14

0.15

0.16

Credit Spread

0 0.1 0.2 0.3 0.4 0.5

c

1.25

1.3

1.35

1.4

1.45

1.5

1.55

1.6

1.65

Investor Payoff

Figure 11: Credit Spreads and Investor Value

Intuitively, this occurs because to incentivize disclosure and prevent diversion P faces

the trade-off of either loading on M’s rents or raising the termination odds, both of which

are costly. When the firm is likely to obtain low cash flows, the chance of terminating the

firm is high and therefore termination is more costly. If the investment cost c > c, the

optimal policy loads more on managerial rents (larger v0). As the cost drops, evidence

is possibly produced and disclosed, which alleviates the termination concern. So the

optimal policy loads on less rents (lower v0).

10 Conclusions

We study the implications of embedding voluntary disclosure of evidence a la Dye (1985)

in an otherwise standard dynamic agency model with non-verifiable cash flows, similar

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to Biais et al. (2007). The model captures three key empirical regularities: (i) techno-

logical progress (as well as regulation) increasingly promotes the use of evidence about

performance; (ii) evidence is decentralized, namely, it is typically better observed and

understood by a firm’s management, than by its arm’s length stakeholders. So, its dis-

closure needs to be costly incentivized; (iii) evidence has a forward looking dimension: it

is useful to predict future cash flows.

The optimal contract can still be implemented by simple securities, as proposed in

DeMarzo and Fishman (2007): the firm borrows both short and long term, and retains

a fraction of its equity. The main difference here is that the interest rate has to depend

on both performance and evidence disclosure; in particular, when bad news are disclosed

the one-period interest charges are forgiven by the principal. In all other circumstances,

the interest rate is higher the more widespread evidence availability is.

We find that the presence of evidence reduces the optimal pay-for-performance sen-

sitivity, because it enables the investors to condition their short-term liquidity prevision

on both the reported cash flows and the evidence produced by the management. If the

managers can convince the investors that a temporary negative performance is due to bad

luck, as opposed to bad behavior, the investors can cut the firm some slack and accept a

temporary relief on interest payments.

While this beneficial effect of evidence reduces the firm’s credit spread in secondary

markets, when no capital structure decisions are made, the result may reverse in primary

markets. Here, both the firm’s initial liquidity and its credit spread might be non-

monotonic functions of disclosure. Namely, better evidence might lower firm value at the

constrained optimal allocation, exacerbating the conflict between rent extraction by the

principal and efficiency. This occurs especially at low profitability firms, because better

evidence reduces the marginal value of providing initially financial slack to the firm, so

that P trades-off higher liquidation odds with a lower managerial pay level.

We characterize the policy dynamics and show that evidence brings about two counter-

vailing dynamic forces. On the one hand, it increases the persistence of a firm’s liquidity

conditional on the firm experiencing poor performance that can be surely attributed to

bad luck (i.e., conditional on bad news being disclosed). On the other hand, by requiring

higher interest rates on the short-term debt, it deteriorates faster the firm’s liquidity

when bad news are not disclosed and performance is poor.

Our numerical simulations suggest that while generating a relatively small increase

in stakeholder’s value, evidence can dramatically reduce efficiency, increasing the liqui-

dation odds and the minimal time required to reach the liquidation boundary, as well as

inducing volatility spikes in continuation utilities for managers and in liquidation odds.

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Importantly, the inefficiency induced by more frequent evidence disclosure that we derive

arises in a model where the principal has full commitment power; it does not depend on

the presence of time inconsistencies such as limited commitment.

We believe that our results might be a useful benchmark both to construct more

complicated models that take seriously the possibility of limited commitment, and em-

pirical exercises that seek to establish the relation between a firm’s liquidity and either

the frequency of its disclosure, or its price impact.

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A Appendix

Proof of Lemma 1. The T = 1 case taught us two facts: (i) whether P has exercised

or not the option, this has no effect on the last period implementable payoffs; (ii) P could

find it optimal to exercise the option at the beginning only if she terminates in the first

period, after a low state is reported and no evidence is disclosed. So, there are only

three policies to consider: TT (P does not exercise the option and terminates in the first

period when x = l), NT (P does not exercise the option and never terminates) and OT

(P exercises the option in the first period and terminates only when when x = l and there

is no disclosure). Wlog we can set all payments when the last cash flow is l to zero.

Under the policy TT , there is one payment to determine: uhh, that is, the payment to

M aftery two successes. The payment satisfies two ICs: (i) at date 2, uhh ≥ δ+uhl = δ; (ii)

at date 1: puhh ≥ δ. It follows that uhh = δ/p; M’s utility at this policy is UM(TT ) = pδ;

P’s utility is UP (TT ) = (1 + p)(l + p∆)− pδ.Under the policy NT , we need to determine two payments: ulh and uhh. While ulh

only satisfies ulh ≥ δ, uhh satisfies both uhh ≥ δ (at date 2) and puhh ≥ δ+ pδ (at date 1,

where we plugged the optimal ulh = δ). It follows that uhh ≥ δ(1 + p)/p; M’s utility at

this policy is UM(NT ) = 2pδ; P’s utility is UP (NT ) = 2(l + p∆)− 2pδ.

Under the policy OT , we need to determine three payments: udlh and uahh (for a ∈{n, d}). However, from the disclosure IC we can see that unhh = udhh := uhh, and so the

problem reduces to solving for uhh and udlh. For similar reasons as before, udlh = δ. As for

uhh, it must be the same as in policy T , because the only feasible deviation from x = h is

to claim that x = l without disclosing evidence. So, uhh = δ/p; M’s utility is UM(OT ) =

pδ(1+(1−p)π); P’s utility is UP (OT ) = (2−(1−p)(1− π))(l+p∆)−pδ(1+(1−p)π)−c.First, when comparing TT and NT we obtain a threshold p such that:

p :=l(1−∆) + δ −

√4l2∆ + (l(∆− 1)− δ)2

−2l∆

If p > p, P strictly prefers TT ; if p < p, P strictly prefers NT ; if p = p, P is indifferent

between the two policies (or any randomization of the two policies).

Second, either P exercises the option at p, or she never does. So, fixing p = p and

comparing UP (NT ) and UP (OT ) yields the threshold c:

c :=λπ

2l2∆

[l2(1 + ∆2) + δ2 + l(1−∆)2δ − (δ + l(1−∆))

√l2(1 + ∆)2 + 2l(1−∆)δ + δ2

]Focusing on c < c, we need to consider two cases. If p > p, we need to compare UP (TT )

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and UP (OT ). We find that UP (OT ) ≥ UP (TT ) if and only if p ≤ p, where:

p :=lπ(∆− 1)− πδ +

√π(4∆(1− λ)(lπ − c) + π(l(1−∆) + δ)2))

2π∆(1− λ)

Finally, if p < p, we need to compare UP (NT ) and UP (OT ). We find that UP (OT ) ≥UP (NT ) if and only if p ≥ p, where:

p :=(1− π)(l(∆− 1)− δ) +

√(l(1−∆ + δ)(1− π))2 + 4(c+ l(1− π))(l∆(1− π) + πδ)

2(l∆(1− π) + πδ))

Note that c enters p under the square-root and has a negative sign, while it enters p only

under the square-root with a positive sign. It follows that ∂p/∂c < 0 and ∂p/∂c > 0.

To proceed with the proofs for the infinite-horizon model, let us first show some

basic properties of the surplus function. Let C be the space of continuous and bounded

functions on the domain R+. Let F := {q ∈ C : 0 ≤ q ≤ s∗} be endowed with the ’sup’

metric where s∗ = (1+r)µr

is the first best surplus. It’s easy to see that F so defined is a

complete metric space. Define the Bellman operator T : F → F as:

Tq(v) = maxθ,π,ui,wi

(1− θ)µ

+1− θ1 + r

{π[pq(wdh) + (1− p)q(wdl)] + (1− π)[pq(wnh) + (1− p)q(wnl)]

}s.t. (PK), (ICg), (ICb), (ICn), (LL)

It’s standard to show that the Bellman operator T : F → F is well defined and the

constraint set is convex. Moreover, we can show the Bellman operator has the following

property.

Lemma A.1. Let F1 = {q ∈ F : q(v) = s∗ for all v ≥ (1+r)pδr}. If q ∈ F1, then T1 ∈ F1.

Proof. Take any q ∈ F1 and v ≥ (1+r)pδr

. Consider the following policy:

θ = π = udl = unl = 0, udh = unh =v

p− δ

r, wi =

(1 + r)pδ

r∀i ∈ H1

It’s easy to check that this policy satisfies all the constraints of the firm’s problem (S).

In addition, under this feasible policy we know

(Tq)(v) ≥ µ+q(v)

1 + r=

(1 + r)µ

r= s∗

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Hence, we must have (Tq)(v) = s∗.

Proposition A.1. The unique fixed point of T , which we call s(v), is increasing, concave,

and satisfies s(v) = s∗ for any v ≥ (1+r)pδr

.

Proof. It is easy to see that T is monotone (whereby q1 ≤ q2 implies Tq1 ≤ Tq2) and

satisfies discounting (wherein T (q+a) = Tq+ δa). Then the Blackwell’s theorem implies

T is a contraction mapping on F and hence has a unique fixed point in F . Let F2 =

{q ∈ F : q(v) is increasing and concave for all v ∈ R+}. It’s standard to show that T

maps from F2 to F2. Combining Lemma A.1, we must have that the unique fixed point

of T lies in F1 ∩ F2.

Proof of Proposition 2. Given the fact that s(v) reaches first best for a large enough

v, we can define v1 = inf{v : s(v) = s∗} and v2 = inf{v : f(v) = s∗}.First, consider the program (S), and let {θ, ui, wi}i∈H1 be the optimal policy at v.

Note that, to achieve first best, all continuation values wi∈H1 must be no less than v1,

and the liquidation probability θ is zero. In addition, from the constraints (ICg),(ICb),

(ICn), and (LL), we must have

(1 + r)udl + wdl ≥ (1 + r)unl + wnl ≥ v1 (A.14)

(1 + r)udh + wdh ≥ (1 + r)unh + wnh ≥ (1 + r)δ + v1 (A.15)

Then (PK) implies v1 ≥ pδ+ v11+r

, or v1 ≥ (1+r)pδr

. Moreover, the conclusion of Proposition

A.1 implies that v1 ≤ (1+r)pδr

. Hence, we must have v1 = (1+r)pδr

.

Now condiser the program (F ). To achieve first best, the investment option must

be never exercised, and moreover, the policies wi ≥ v2. So similarly, we can show that

v2 ≥ (1+r)pδr

. In addition, since f(v) ≥ s(v; 0), Proposition A.1 implies that v2 ≤ (1+r)pδr

.

Therefore, we must have v2 = (1+r)pδr

.

To characterize the policies, we specify the first order conditions and the envelope

condition of program (S) as follows. Denote η as the Lagrangian multiplier of (PK).

Moreover, let αg, αb, αn be the multipliers of (ICg), (ICb), and (ICn), respectively. Then

the first order conditions are:

(1− θ)πps′(wdh) = (1− θ)πpη − αg (FOCdh)

(1− θ)π(1− p)s′(wdl) = (1− θ)π(1− p)η − αb (FOCdl)

(1− θ)(1− π)ps′(wnh) = (1− θ)(1− π)pη + αg − αn (FOCnh)

(1− θ)(1− π)(1− p)s′(wnl) = (1− θ)(1− π)(1− p)η + αb + αn (FOCnl)

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The envelope condition is:

s′(v) = η (EN)

Proof of Lemma 1. Take any v < v, and let {θ, π, wi∈H1} be the optimal policies of

the program (S) with π = π.

First, we show that αb = 0. Suppose not. Then by the first oder conditions (FOCdl)

and (FOCnl), we must have s′(wdl) < s′(wnl), which further implies that wdl > wnl by

the concavity of s(·). In other words, the constraint (ICb) holds as strict inequality. The

complementary slackness then implies αb = 0, a contradiction.

Second, we show that it cannot be the case that αg = αn = 0. Suppose this is true.

Then all the incentive constraints are not binding. Therefore, the surplus s(v) should be

the same as if we solve the firm’s problem (S) with only the promise keeping constraint

(PK). Since wi = (1 + r)v is feasible, we know from the objective of (S) that s(v) ≥ s∗,

a contradiction with v < v.

Third, we show that αn > 0. Suppose not. Then from the above result, it has to be

that αg > 0 = αn. Then the first oder conditions (FOCnh) and (FOCnl) together imply

that s′(wnh) > s′(wnl). Hence, wnh < wnl, contradicting with (ICn).

Fourth, we show that αg > 0. Suppose not. Then by (FOCdh) and (FOCnh), we

know s′(wdh) > s′(wnh) which further implies that wdh < wnh. This forms a contradiction

with (ICg).

Last, using the facts of αn > 0 = αb, we can conclude from (FOCdl), (FOCnl), and

(EN) that s′(wdl) = s′(v) < s′(wnl). Hence, concavity implies wdl > wnl.

Proof of Proposition 3. The proof is divided into two parts. Part (a) shows that the

optimal policies described in the Proposition satisfy all the constraints and the optimality

conditions of the firm’s problem (S). Part (b) derives the n-period liquidation thresholds.

Part (a). Take any v ≤ v, and let {θ, wh, wdl, wnl} be the optimal policies specified

in the proposition. It’s easy to verify that they satisfy all the constraints of (S). Using

the result of Lemma 1, the first order conditions (FOCdh) to (FOCnl), and the envelop

condition (EN), we can eliminate all the Lagrange multipliers and obtain

[1− (1− p)π]s′(v) = ps′(wh) + (1− p)(1− π)s′(wnl) (A.16)

This optimality condition along with (PK) and (ICn) jointly determine the optimal

policies. Hence, we are left to show that (A.16) always holds under the proposed policies.

First, consider the case of v ≥ v1. In this region, θ = 0. Plug in wh, wdl, wnl to the

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objective of (S) and then differentiate it with respect to v to obtain

(1 + r)s′(v) = p(1 + r)s′(wh) + (1− p)πs′(v) + (1− p)(1− π)(1 + r)s′(wnl) (A.17)

Then plug in the expression r = r1−(1−p)π to (A.17) and rearrange will give exactly (A.16).

Second, in the case of v < v1, we can plug in θ, wh, wdl, wnl to the objective of (S) to

obtain s′(v) = s(v1)v1

= s′(v1). Since wh, wnl do not vary with continuation utility when it

moves from v1 to v and since (A.16) holds at v1, it must also hold at v.

Part (b). Notice that at the n-period thresholds the following must hold: wnl(v1) = 0,

and wnl(vj) = vj−1 for j ≥ 2. According to the optimal policy of wnl, the latter implies

vj−1 = wnl(vj) = vj − r(v − vj)

Hence, vj = 11+r

[vj−1 + rv]. Moreover, by the optimal policies and (PK), we have

(1 + r)v1 = p(1 + r)δ + (1− p)πv1 = rv + (1− p)πv1

which implies v1 = r1+r

v. Finally, we can obtain the threshold expression simply by

induction.

Proof of Proposition 4. Part (a). Consider any π < 1 as a parameter of the firm’s

problem (S). Take any continuation value v at the no-liquidation region [v1, v). Let

wdl, wnl be the optimal policies at v. Denote sπ(v) to be the derivative of s with respect

to π at the fixed value v. Then the envelop condition with respect to π is

(1 + r)sπ(v)

1− p= s(wdl)− s(wnl)− s′(wdl)(wdl − wnl) (A.18)

Since s(·) is concave, wnl < wdl, and s′(wdl) < s′(wnl) (see the last part of the Lemma

1 proof), we must have sπ(v) > 0. In addition, since s(·) is linear in v for any v < v1,

continuity of s(·) implies that sπ(v) > 0 for v < v(1).

Part(b). See the proof in Proposition 2.

Part(c). Take any v ∈ [v1, v] (no-liquidation region). According to the definition in

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(5), the pay-performance sensitivity can be calculated as

PPS =wh + (1 + r)uh − [πwdl + (1− π)wnl]

(1 + r)∆

=πwnl + (1 + r)δ − πwdl

(1 + r)∆

=π[v − r(v − v)] + (1 + r)δ − πv

(1 + r)∆

= λ− πr(v − v)

(1 + r)∆

The second line is from the fact of (ICn) being equality, while the third line is from

plugging in the policy expressions of wdl, wnl. Obviously, PPS is decreasing in π and

increasing in v.

Now consider the liquidation region. Since our PPS measure is only defined when the

firm is not liquidated in the beginning of the period, we can simply replace the v in the

above derivation by v(1) and get

PPS = λ− πr(v − v1)

(1 + r)∆=

λ(1 + r − π)

1− (1− p)π + r

Obviously, in this case, PPS does not depend on v and decreases in π.

Proof of Proposition 5. For v ∈ (0, v), since π > 0 and Proposition 4 shows that s(v)

strictly increases in π, we must have s(v; π) > s(v; 0). Then the continuity of s(·) implies

c in (8) is well defined, and c > 0. Moreover, the threshold cost c increases in π because

s(v; π) does.

When π increases, note that the increase of s(v; π) is lager than that of f(v). This is

becasue by the problem (F ) the increase of f(v) is due to future increase in firm value

when the option is exercised. Hence, vh becomes larger and vl beomes smaller.

When c increases, similar argument implies that the decrease of s(v; π) is lager than

that of f(v). Hence, vh becomes smaller and vl beomes larger.

Proof of Proposition 6. If c = 0, then the option is exercised right away because

s(v; π) ≥ s(v; 0) for any v. In this case, vl = 0 and vh = v.

Now consider any 0 < c < c. First, not that in the region close to the boundary v,

not exercising the option is optimal. This is because f(v) = s∗ > s(v; π)− c. Then vh as

in (9) is well defined, and the option is not exercised for v ∈ [vh, v].

Second, not exercising the option is optimal in the region close to the boundary 0.

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This is becasue f(0) = 0 > s(0; π)− c. Hence, vl as in (9) is well defined, and the option

is not exercised for v ∈ [0, vl].

Third, suppose the option is also delayed for v ∈ (vl, vh). Then the option is never

exercised, and f(v) = s(v; 0) for all v ∈ [0, v]. However, since c < c, there are some v

such that s(v; π)− c > s(v; 0) = f(v), implying exercise the option is optimal at such v.

This is contradiction.

In the case of c > c, we have s(v; π) − c < s(v; 0) for any v, by the definition of c in

(8). Then s(v; π)− c < f(v) for any v, implying the option is never exercised.

Lemma A.2. When the investment option is not exercised, the relevant optimal contin-

uation utility wh and wnl are the ones in Proposition 3 by setting r = r.

Proof. Take any v < v. We first show that the (ICn) constraint in problem (F ) must

hold as equality at v. Suppose not. Then f(v) should be the same as if we solve (F )

without (ICn). Then since wnh = wnl = (1 + r)v and θ = 0 is feasible, we know from the

objective of (F ) that f(v) ≥ s∗, a contradiction with v < v.

Then the optimal policy at v is jointly determined by (PK) and the equality of (ICn),

resultin in the expressions in Proposition 3 with π = 0.

Proof of Proposition 7. We will show that under the arrangement implements the

payout and the evolution of the optimal contract.

First, in the cash payout region the balance of credit line becomes zero or m = 0. The

agent’s total payoff is λ fraction of the firm payout with is λdh = uh. And according to

(11) (12) (13), the subsequent credit line balance will all be zero, or mh = mnl = mdl = 0,

which implis wh = wnl = wdl = v by (10). So in the implementation, the agent gets the

same cash as in the contract, and the firm always stays at v, having no probability of

defaulting.

Second, in the none-cash-payout region, given the current credit balance is m, we can

use (10) to (13) to derive the subsequent credit line balance as:

m = mdl =v − vλ

mnl =[1 + r(π)](v − v)

λ, mh = mnl − (1 + r)∆

So by (10), M’s continuation utility (by withdrawing all available credit) becomes

wdl = v, wnl = v − r(π)(v − v), wh = wnl + (1 + r)δ

which is the same as in Proposition 3. Note this derivation includes the both senarios of

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whether the investment option is exercised or not. In the latter case, we only have the

subsequent credit balance to be mnl or mh, and the continuation utility to be wnl or wh.

These values are obtained by setting π = 0 in the above expressions. Hence, by Lemma

A.2 we know these continuation utilities are the same as in the option contract.

Lemma A.3. In the case of p ≤ r, we have s(v) = ai + biv for any v ∈ [vi, vi+1n], where

i = 0, 1, 2..., and the coefficients satisfy

a0 = 0, bi =µ(r + p)(1 + r)

rpδ[r(1− p)(1− π)/r]i (A.19)

Proof. According to Proposition 3, when p ≤ r we will have wh(v) ≥ v for any v > 0.

In other words, the firm will immediately reach the payout boundary v after any high

cash-flow shock, conditional on the firm is not liquidated in the beginning of the period.

From this observation, we can derive s(·) by induction.

When v ∈ (0, v1], the objective of (A.1) implies

s(v) =v

v1

{µ+

1

1 + r[ps∗ + (1− p)πs(v)]

}

from which we get s(v1) = (1+r)(r+p)µr[1+r−(1−p)π]

. Hence, b0 = s(v1)v1

= µ(r+p)(1+r)rpδ

.

When v ∈ [vi, vi+1] for i ≥ 1, we have wnl(v) = (1 + r)v − rv. Then the objective of

(A.1) implies

[1 + r − (1− p)π]s(v) = (1 + r)µ+ ps∗ + (1− p)(1− π)s(wnl)

and therefore

ai + biv =(1 + r)µ+ ps∗

1 + r − (1− p)π+

(1− p)(1− π)

1 + r − (1− p)π{ai−1 + bi−1[(1 + r)v − rv]}

Equating the coefficients, we get bi = rr(1− p)(1− π)bi−1. The result is then obtained by

induction.

According to (A.19), the slope bi is a function of p and π. We denote it as bi(π; p),

and denote the derivative of bi with respect to π as b′i(π; p).

Lemma A.4. Take any i ≥ 1. There exists p ≤ r such that b′i(0; p) > 0. Moreover,

b′i(0; p = r) < 0.

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Proof. From the expression of bi in (A.19), we get

b′i(0; p) =µ(r + p)

rpδ(1− p)i[(1− p)r − i(1 + r)p]

Clearly, b′i(0; p) > 0 if p is close to zero, while b′i(0; p = r) < 0.

Lemma A.5. There exists some j ≥ 1 and p = r = r, such that bj(0) = 1.

Proof. From the expression of bi in (A.19), we get

bi(0; p = r) =µ(r + p)

rpδ(1− p)i

= 2(

1 +l

r∆

)(1 +

1

r

)(1− r)i

Clearly, bi(0; p = r) decreases in r and decreases in i. And b0(0; p = r) > 2 and b∞(0; p =

r) = 0. Hence, there must exist j and r such that bi(0; p = r) = 1.

Proof of Proposition 8. Note that if c = c the firm never exercises its investment

option, and at initiation, the investors choose

v0 = arg maxv{s(v; 0)− v}

Consider the values j and r in Lemma A.5, and let p = r = r. Then we must have

s′(v0; 0) = 1, and v0 = vj+1. Given the resul t of Lemma A.4, we know there exists π > 0

such that bj(π; r) < 1 for any π < π.

On the contrary, if c = 0, the firm exercises the option at initiation. Hence,

v0 = arg maxv{s(v; π)− v}

If π is sufficiently small and π < π, the above argument implies that v0 ≤ vj < v0. And

because s(v; π) is continuous in both v and π. A small change in π and a downward jump

of initial continuatin utility from v0 to v0 implies s(v0; π) < s(v0; 0). So the firm value

and M’s rent at initiation must both have a increasing part in c.

Now conside the case where p is close to zero. Lemma A.4 implies s′(v0; 0) < s′(v0; π)

for a sufficiently small π. Hence, v0 > v0, which further implies s(v0; π) > s(v0; 0), since

s(·; π) is increasing in π. Therefore, the firm value and M’s rent at initiation must both

have a decreasing part in c.

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